14
Modelowanie Nanostruktur Lecture 4 1 Modelowanie Nanostruktur Semester Zimowy 2011/2012 Wykład Jacek A. Majewski Chair of Condensed Matter Physics Institute of Theoretical Physics Faculty of Physics, Universityof Warsaw E-mail: [email protected] Modelowanie Nanostruktur, 2011/2012 Jacek A. Majewski Wykład 4 25 X 2011 Continuous Methods for Modeling Electronic Structure of Nanostructures Effective Mass Approximation Modeling Nanostructures Nanotechnology Low Dimensional Structures Quantum Wells Quantum Wires Quantum Dots A B Simple heterostructure

Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

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Page 1: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 1

Modelowanie Nanostruktur

Semester Zimowy 2011/2012

Wykład

Jacek A. Majewski

Chair of Condensed Matter Physics

Institute of Theoretical Physics

Faculty of Physics, Universityof Warsaw

E-mail: [email protected]

Modelowanie Nanostruktur, 2011/2012

Jacek A. Majewski

Wykład 4 – 25 X 2011

Continuous Methods for Modeling Electronic Structure of Nanostructures Effective Mass Approximation

Modeling Nanostructures

Nanotechnology –

Low Dimensional Structures

Quantum

Wells

Quantum

Wires

Quantum

Dots

A B

Simple

heterostructure

Page 2: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 2

What about realistic nanostructures ?

2D (quantum wells): 10-100 atoms in the unit cell

1D (quantum wires): 1 K-10 K atoms in the unit cell

0D (quantum dots): 100K-1000 K atoms in the unit cell

Organics

Nanotubes, DNA: 100-1000 atoms (or more)

Inorganics

3D (bulks) : 1-10 atoms in the unit cell

TEM image of a InAs/GaAs dot Si(111)7×7 Surface

GaN

InGaN

GaN

HRTEM image:

segregation of Indium

in GaN/InGaN

Quantum Well

Examples of Nanostructures

Synthesis of colloidal nanocrystals

Injection of organometallic

precursors

Mixture of surfactants Heating mantle

Nanostructures: colloidal crystals

-Crystal from sub-µm

spheres of PMMA

(perpex) suspended in

organic solvent;

- self-assembly when

spheres density high

enough;

illuminated with white light

Bragg„s law different

crystals different orientation

different

Page 3: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 3

Hot topic (to come) –

The curious world of nanowires

Self-organized growth of nanowires:

catalytic VLS growth

Mechanizm Vapor-Liquid-Solid

VLS (Wagner 1964)

SEM of ZnTe nanowires grown by MBE

on GaAs with Au nanocatalyst

Nanowire site control and branched NW structures:

nanotrees and nanoforests

(A) Nanowires can be accurately positioned using lithographic methods such as EBL and NIL. (B) Subsequent seeding by aerosol

deposition produces nanowire branches on an array as in panel (A). Shown here is a top view of such a ‘nanoforest’ where the

branches grow in the <111>B crystal directions out from the stems. (C) Dark field STEM image and EDX line scan of an

individual nanotree. An optically active heterosegment of GaAsP in GaP has been incorporated into the branches.

http://www.nano.ftf.lth.se/

Page 4: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 4

Nanowire nanolasers

Room temperature lasing action from chemically synthesized

ZnO nanowires on sapphire substrate

Huang, M., Mao, S.S., et al., Science 292,

1897 (2001)

Schematic illustration A SEM image

One end of the nanowire is the epitaxial interface between the

sapphire

and ZnO, whereas the other end is the crystalline ZnO (0001)

plane

exposed in air.

Detection of single viruses with NW-FET

Patolsky & Lieber, Materials Today, April 2005, p. 20 Patolsky et al., Proc. Natl. Acad. Sci.

USA 101 (2004) 14017

Simultaneous conductance and optical data recorded

for a Si nanowire device after the introduction of influenza A solution

Controlled Growth and Structures of

Molecular-Scale Silicon Nanowires

(a) TEM images

of 3.8-nm SiNWs

grown along the

<110> direction

(c) cross-sectional

image

(b) & (d) models

based on

Wulff

construction

Yue Wu et al., NANO LETTERS 4, 433

(2004)

High Performance Silicon Nanowire

Field Effect Transistors

Yi Cui, et al. NANO LETTERS 3, 149

(2003)

Comparison of SiNW FET

transport parameters

with those for

state-of-the-art planar

MOSFETs show that

“SiNWs have the

potential to exceed

substantially conventional

devices, and thus could be

ideal building blocks for future

nanoelectronics.”

Page 5: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 5

Heterostructured Nanowires

coaxial

heterostructured

nanowire

longitudinal

heterostructured

nanowire

Heterostructured Nanowires

Transmission electron microscopy images of

a GaN/AlGaN

core–sheath nanowire

two Si/SiGe

superlattice nanowires

Simulation methods

Atomistic methods for modeling of

nanostructures

Ab initio methods (up to few hundred atoms)

Page 6: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 6

Density Functional Theory (DFT)

One particle density determines the ground state energy

of the system for arbitrary external potential

extE [ ρ] d r ρ( r )υ ( r ) F [ ρ] 3

E [ ρ ] E0 0

ground state density

ground state energy

ext S x cE[ ρ] drυ ( r )ρ( r ) T [ ρ] U [ ρ] E [ ρ] E [ ρ]

unknown!!!

Total energy

functional

External

energy

Kinetic

energy

Classic Coulomb

energy

Exchange

energy

Correlatio

n

energy

Atomistic methods for modeling of

nanostructures

Ab initio methods (up to few hundred atoms)

Semiempirical methods (up to 1M atoms)

Tight-Binding Methods

Tight-Binding Hamiltonian

† †iα iα iα iα , jβ iα jβ

αi αi ,βj

H ε c c t c c

creation & anihilation operators

On-site energies are not atomic eigenenergies

They include on average the effects

of neighbors

Problem: Transferability

E.g., Si in diamond lattice (4 nearest neighbors)

& in fcc lattice (12 nearest neighbors)

Dependence of the hopping energies on the distance

between atoms

0 2 6 4 8 10 12 14 1

10

100

1 000

10 000

100 000

1e+06

Nu

mb

er

of

ato

ms

R (nm)

Tight-Binding

Pseudo-

potential

Ab initio

Atomistic vs. Continuous Methods

Microscopic approaches can be applied

to calculate properties of realistic nanostructures

Number of atoms in a spherical Si nanocrystal as a function of its radius R.

Current limits of the main techniques for calculating electronic structure.

Nanostructures commonly studied experimentally

lie in the size range 2-15 nm.

Continuous

methods

Page 7: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 7

Atomistic methods for modeling of

nanostructures

Ab initio methods (up to few hundred atoms)

Semiempirical methods (up to 1M atoms)

(Empirical Pseudopotential)

Tight-Binding Methods

Continuum Methods

(e.g., effective mass approximation)

Continuum theory-

Envelope Function Theory

Electron in an external field

2ˆ( ) ( ) ( ) ( )

2

pV r U r r r

m

Periodic potential of crystal Non-periodic external potential

Strongly varying on atomic scale Slowly varying on atomic scale

0

-5

5

1

3

1

1

1

3

„2

„25

15

1

„2

„2

5

1

5

L „2

L1

L „3

3L

L14

1

En

erg

y [

eV

]

Wave vector k

11

Ge

Band structure

of Germanium

( )n

k0U( r )

Band Structure

Electron in an external field 2ˆ

( ) ( ) ( ) ( )2

pV r U r r r

m

Periodic potential of crystal Non-periodic external potential

Strongly varying on atomic scale Slowly varying on atomic scale

Which external fields ?

Shallow impurities, e.g., donors

Magnetic field B,

Heterostructures, Quantum Wells, Quantum wires, Q. Dots

2

( )| |

eU r

r

B curlA A

GaAs GaAlAs

cbb

GaAs GaAlAs GaAlAs

Does equation that involves the effective mass and a slowly varying

function exist ? 2ˆ( ) ( ) ( )

2 *

pU r F r F r

m

( ) ?F r

Page 8: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 8

Envelope Function Theory –

Effective Mass Equation J. M. Luttinger & W. Kohn, Phys. Rev. B 97, 869 (1955).

[ ( ) ( ) ] ( ) 0ni U r F r

0( ) ( ) ( )n nr F r u r

( ) 0U r ( ) exp( )nF r ik r

(EME)

EME does not couple different bands

Envelope

Function

Periodic

Bloch Function

“True”

wavefunction

Special case of constant (or zero) external potential

( )r Bloch function

( )U z ( ) exp[ ( )] ( )n x y nF r i k x k y F z

Electronic states in

Quantum Wells

Envelope Function Theory- Electrons in Quantum Wells

Effect of Quantum Confinement on Electrons

Let us consider an electron in the conduction band near point

GaAs GaAlAs GaAlAs

cbb

Growth direction (z – direction )

Potential ? ( )U r

( )U r is constant in the xy plane

22

0 *( )

2c ck k

m

GaAs

GaAlAs

0

0

( ) ( )c

c c

zU r U z

E z

2 2 2 2

2 2 2( , , ) ( ) ( , , ) ( , , )

2 *F x y z U z F x y z EF x y z

m x y z

Effective Mass Equation for the Envelope Function F

( , , ) ( ) ( ) ( )x y zF x y z F x F y F zSeparation Ansatz

2 222

2 2 2( )

2 *

y zxy z x z x y x y z x y z

F FFF F F F F F U z F F F EF F F

m x y z

2 222

2 2 2( )

2 *

y zxy z x z x y x y z x y z

F FFF F F F F F U z F F F EF F F

m x y z

x y zE E E E

22

22 *

xy z x x y z

FF F E F F F

m x

22

22 *

yx z y x y z

FF F E F F F

m y

22

2( )

2 *

zx y x y z z x y z

FF F U z F F F E F F F

m z

22 2

2

2~ ,

2 * 2 *xik xx

x x x x x

FE F F e E k

m mx

Effective Mass Equation of an Electron in a Quantum Well

22 22

2~ ,

2 * 2 *

yik yyy y y y y

FE F F e E k

m my

22

2( )

2 *

znzn zn zn

FU z F E F

m z

Page 9: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 9

Conduction band states of a Quantum Well

|||| || ||

1 1( ) exp[ ( )] exp( )x yk

F r i k x k y ik rA A

|| |||| || ||,

1( ) ( ) ( ) exp( ) ( )zn znn k k

F r F r F z ik r F zA

22

|| ||( )2 *

n znE k k Em

Energies of bound states

in Quantum Well

22

2( )

2 *

znzn zn zn

FU z F E F

m z

En

erg

y

n=1

n=2

n=3

0c

0c cE znE ||( )nE k

||k

22

1 ||2 *

E E km

22

2 ||2 *

E E km

1E

2E

Wave functions Fzn(z)

znE

( )znF z

Band structure Engineering

The major goal of the fabrication of heterostructures is the

controllable modification of the energy bands of carriers

Lattice constant

En

erg

y G

ap

[e

V]

Band lineups in semiconductors

-12

-10

-8

-6

-4

0

InS

b

GaS

b

InA

s

AlS

b

GaA

s

AlA

s

InP

GaP

AlP

InN

GaN

AlN

Zn

O

Zn

S

Cd

S

Zn

Se

EN

ER

GY

(e

V)

Band edges compile by: Van de Walle (UCSB) Neugebauer (Padeborn), Nature’04

e.g., GaAs/GaAlAs InAs/GaSb

Band lineup in GaAs / GaAlAs

Quantum Well with Al mole fraction

equal 20%

Envelope Function Theory- Electrons in Quantum Structures

Type-I Type-II

vbt

cbb A

B

vE

cE

AgapE

BgapE

vbt

cbb

A B

vE

cE

AgapE

BgapE

vbt

cbb A

B vE

cE

AgapE

BgapE

Various possible band-edge lineups in heterostructures

GaN/SiC

(staggered) (misaligned)

- Valence Band Offset (VBO) vE - Conduction band offset cE

GaAlAs vbt

cbb

1.75 eV 1.52 eV

0.14 eV

0.09 eV

GaAs

VBO’s can be only obtained either from experiment or ab-initio calculations

Page 10: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 10

Density of states

We define to refer to a single direction of electron spin

(denoted )

Density of States

G(E)dE - the number of electron states with energies E and E + dE

Generally, this quantity is proportional to the volume of the crystal

Convention: we define G to refer to the volume of a unit cell

3

3( ) ( ( ))

(2 )n

n

G E d k E k

G

Of course, there is no contribution from

a particular band at energy E unless

there are states of energy E in that band

Band index

2 2

*( ) (0)

2n n

n

kk

m

3/ 2(3 )

2 2

* 2( )

D mG E E

Density of States of a Two-Dimensional Electron Gas

A special function known as the density of states G(E) that gives

the number of quantum states dN(E) in a small interval dE around

energy E: dN(E)= G(E) dE

-the set of quantum numbers (discrete and continuous)

corresponding to a certain quantum state

( ) ( )G E E E

Energy associated with

the quantum state

||{ , , }s n k For 2DEG:

Spin quantum number Continuous two-dimensional vector

A quantum number characterizing the

transverse quantization of the electron states

22 2

, ,

( ) 2 [ ( )]2 *

x y

n x y

n k k

G E E E k km

Density of States of a Two-Dimensional Electron Gas 2

2 2

, ,

( ) 2 [ ( )]2 *

x y

n x y

n k k

G E E E k km

,x yL L - are the sizes of the system in x and y directions

x yS L L - the surface of the system 2

,

( ) ( )(2 )

x y

x yx y

k k

L Ldk dk

22 2

2

22

|| || ||2 0

2|| || ||2 0

( ) 2 [ ( )]2 *(2 )

2 ( )2 *2

2 *( )

x yx y n x y

n

x yn

n

x yn

n

L LG E dk dk E E k k

m

L Lk dk E E k

m

L L mk dk E E k

2|| ||k

|| ||2 20

* *( ) ( ) ( )n n

n n

Sm SmG E d E E E E

( )x - Heaviside step function for and for ( ) 1 0 ( ) 0 0x x x x

Page 11: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 11

Density of States of a Two-Dimensional Electron Gas

2

*( ) ( )n

n

SmG E E E

Often the density of states per unit area,

, is used to eliminate the size

of the sample

( ) /G E S

Each term in the sum corresponds to the contribution from one subband.

The contributions of all subbands are equal and independent of energy.

The DOS of 2DEG exhibits a staircase-shaped energy dependence,

with each step being associated with one of the energy states.

De

ns

ity o

f s

tate

s

1E 2E 3E E

(2 )( )

DG E

3/ 2(3 )

2 2

* 2( )

D mG E E

(3 )( )

DG E

Density of states for 2DEG in an

infinitely deep potential well

22

*( )

2k k

m

For large n, the staircase

function lies very close to

the bulk curve (3 )

( )D

G E

2

*m

2

2 *m

2

3 *m

Density of States of a Two-Dimensional Electron Gas

Bulk

300 A QW 100 A QW

0 100 200

De

ns

ity o

f s

tate

s

Energy [meV]

Dependence on the width of Quantum Well

QW of large width Bulk

Electronic states in

Quantum Wires

Electron States in Quantum Wires

xk

Free movement in the x-direction,

Confinement in the y, z directions

( , , ) ( , )xik xnF x y z e F y z

2 2 2

2 2( , ) ( , ) ( , ) ( , )

2 *n n n nF y z U y z F y z E F y z

m y z

22

( )2 *

n x n xE k E km

2 2(1 )

,

( ) 2 ( )2 *

x

D xn

n k

kG E E E

m

(1 )

2

2 * 1( ) ( )

D xn

n n

L mG E E E

E E

( , )U y zConfinement potential

Density of states for one-dimensional electrons

De

ns

ity o

f s

tate

s

1E 2E 3E E

Page 12: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 12

Electronic states in

Quantum Dots

Electron States in Quantum Dots

A

B

A

Self-organized quantum dots

Electrons confined

in all directions

( , , )U x y z

2 2 2 2

2 2 2( , , ) ( , , ) ( , , ) ( , , )

2 *n n n nF x y z U x y z F x y z E F x y z

m x y z

(0 )( ) ( )

DG E E E

Density of states for zero dimensional (0D) electrons (artificial atoms)

De

nsity o

f sta

tes

1E 2E 3E E4E

Confinement Quantization of energy levels

L

L

L

How small should be semiconductor system to see

separate levels in room temperature?

x x

y y

z z

k L πn

k L πn

k L πn

2

2

2

x y z

πE )

m* L( n n n

22 2 22

2

2

ΔE

BΔE k K . eV 300 0 026

m* m* m 0

nmL .m*

2 21173 55

GaAs m* . 0 03 L nm 76

Si m* . 0 2 L nm 29

Electronic states in

Nanostructures

Summary

Page 13: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 13

Energy spectrum

1D: free motion in x direction, dimensional quantisation in z,y

[H,px] = 0

m,n(r) = uk(r)eikxxFm,n(y,z)/(Lx)1/2

2D: free motion in x,y directions,dimensional quantisation in z

[H,px] = 0; [H,py] = 0

n(r) = uk(r)ei(kxx+kyy)Fn(z)/(Lx Ly)1/2

3D: free motion in x,y,z directions

[H,p] = 0

k(r) = uk(r)eikr /(Lx Ly Lz)1/2

Density of States of Electrons in

Semiconductor Quantum Structures

A A B Quantum Wells

A B A

B Bulk

Quantum Wires

Quantum Dots

3D

2D

1D

0D

(2 )( )

DG E

1E 2E 3E E

2

*m

2

2 *m

2

3 *m

(3 )( )

DG E

E

1E 2E 3E E

(1 )( )

DG E

1E 2E 3E E4E

(0 )( )

DG E

3/ 2(3 )

2 2

* 2( )

D mG E E

(2 )

2

*( ) ( )

Dn

n

mG E E E

(0 )( ) ( )

Dn

n

G E E E

(1 )

2 2

( )2 *( )

D n

n n

E EmG E

E E

22

0 *( )

2c ck k

m

B A

A

Position Dependent

Effective Mass

Effective Mass Theory with Position Dependent

Electron Effective Mass

* *A Bm m

*Am *

Bm

0z

2 2

22 * ( )

d

m z dz

21

2 * ( )

d d

dz m z dz

21

( ) ( ) ( ) ( )2 * ( )

d dF z U z F z EF z

dz m z dz

ˆ ˆ[ * ( )] [ * ( )] [ * ( )]z zT m z p m z p m z 2 1

( )F z1 ( )

* ( )

dF z

m z dz

0z

*Am

*( )Bm z

“Graded structures”

IS NOT HERMITIAN !!

Symetrization of the kinetic energy operator

General form of the kinetic energy operator

with

IS HERMITIAN !

and ARE CONTINOUOS !

Page 14: Semester Zimowy 2011/2012 Wykład Modeling Nanostructuresjmajewsk/MODNanoSTR_L4.pdf · Modelowanie Nanostruktur Lecture 4 2 What about realistic nanostructures ? 2D (quantum wells):

Modelowanie Nanostruktur

Lecture 4 14

Doping in Semiconductor

Low Dimensional Structures

vE

cE

Energy band diagram of a selectively doped AlGAAs/GaAs

Heterostructure before (left) and after (right) charge transfer

AlGaAs

GaAs

VACUUM LEVEL

vE

EF

AgE

BgEA

B

AgE

1

dl

ALNegatively

charged

region

Positively charged

region

A Band - The electron affinities of material A & B

The Fermi level in the GaAlAs material is supposed to be pinned

on the donor level.

The narrow bandgap material GaAs is slightly p doped.

BgE

(0)( ) ( ) ( )U z U z e z

Effects of Doping on Electron States in Heterostructures

+ +

Ec

+ + EF + + +

+ + + Ec (z) EF

1E

Unstable Thermal equilibrium Charge transfer

2 2

( ) ( )

4( ) ( ) ( ) | ( ) |A D

acc don

er r R r R f r

Resulting electrostatic potential

should be taken into account in the Effective Mass Equation

2 2 2 2

2 2 2( , , ) ( ) ( ) ( )

2 *U x y z e r r E r

m x y z

2 24( ) ( ) ( ) | ( ) |A D

er N r N r f r

Electrostatic potential can be obtained from the averaged acceptor and

donor concentrations

Fermi distribution function

The self-consistent problem, so-called “Schrödinger-Poisson” problem

Thank you!